GFO Part I Basic Principles
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2.4 Translations and Ontological Mappings

The comparison of ontologies assumes a notion of semantic transformation and ontological mapping.

2.4.1 Semantic Translations

Let $\mathcal{O}=(L,V,Ax)$ be an ontology and $V \Lsubseteq V'$; we say that a sentence $\phi$ from $L(V')$ is ontologically compatible with $\mathcal{O}$ if $\phi$ is consistent with $Ax$. A semantic mapping (or semantic transformation) of an ontology $\mathcal{O}_{1} = (L_{1}, V_{1},
Ax_{1})$ into the ontology $\mathcal{O}_{2} = (L_{2}, V_{2}, Ax_{2})$ is a computable function $f: L_{1} \to L_{2}$ such that $Ax_{2}
\models f(Ax_{1})$. The most important semantic mappings are interpretations in the sense of logic and model theory (59).

We sketch the main ideas of the method of interpretability in the framework of theories in first-order logic (cf. (58)). A theory $S$ is said to be interpretable in the theory $T$ if it is obtainable from $T$ by means of definitions. The question is which schemas of definitions are admitted, and what - in general - a definition is. The simplest case of definitions are explicit definitions which are assumed in the sequel. Let us assume that $S$ and $T$ are theories in the (first-order) languages $L(V)$, and $L(W)$, respectively. Translations from $L(V)$ into $L(W)$ are defined by means of codes. A code in the sense of (58) - in the simplest case - has the form $c = (1, U(x), F_{1}, {\ldots}, F_{n})$, where $U$, $F_{1}$, ..., $F_{n}$ are formulas of the language $L(W)$ specified in the vocabulary $W$; here, a formula $F_{i}$ is associated to every relation symbol $r_{i} \Lin V$, such that the arity of $r_{i}$ equals the number of free variables of $F_{i}$. The formulas $F_{i}$ serve as explicit definitions of the relational symbols $r_{i}$. A translation $tr$ from $L(V)$ into $L(W)$ associates to every formula of $L(V)$ a formula of $L(W)$. Translations based on a code $c$ are recursively defined (for details, see (58)).

A theory $S$ is said to be (syntactically) $c$-interpretable in $T$ if $tr$ - which is based on the code $c$ - satisfies the following condition:

(C) For every sentence $\phi \Lin L(V)$ holds: $S \models \phi$ if and only if $T \models tr(\phi)$.

Generally, a theory $S$ is interpretable in $T$ if a code $c$ exists such that the translation $tr$ which is based on $c$ satisfies condition (C). Note that codes can be much more complicated than the simple version mentioned above.

2.4.2 Ontological Mappings and Reductions

In modelling a concrete domain $D$ we start with a body of source information about $D$, denoted by $\GSI(D)$, which is usually presented in different languages (including natural language), often in a non-structured form. From $\GSI(D)$ a specification $\GSpec(SI)$ (which takes the form of a set of expressions) is constructed with the aim to capture the knowledge-content of $\GSI(D)$. Usually, $\GSpec(SI)$ is expressed in a (formal) modeling/representation language, but also in natural or semi-formal languages, here denoted by ML (modeling language). Examples of such languages are: KIF (22), Description Logics (4), Conceptual Graphs (53), and Semantic Networks, but also modeling languages like UML (Unified Modeling Language) (48) or OPM (Object Process Methodology) (20). In general, the system $\GSpec(SI)$ is not sufficiently ontologically founded, and it remains the task to translate it into an ontologically founded and formal knowledge base which is formulated in some target language $\GOL$ (Ontology Language). An ontological mapping translates the expressions of $\GSpec(SI)$ into the language $\GOL$ resulting in the knowledge base $\GOKB(\GSpec(SI))$, which captures formally the ontological content of $\GSpec(SI)$. We say, in this case, that $\GOKB(\GSpec(SI))$ is an ontological foundation of $\GSpec(SI)$.

We explain the notion of ontological mapping for terminology systems. In general, a terminology system $\mathcal{T} = (L, \GConc, \GRel, \GDef)$ consists of a language $L$, a set $\GConc$ of concepts, a set $\GRel$ of relations between these concepts, and a function which associates to every concept or relation $c \Lin \GConc \cup \GRel$ a definition $\GDef(c)$ which is an expression of the language $L$.

Let $\mathcal{T} = (L, \GConc, \GRel, \GDef)$ be a terminology system and $\mathcal{O}= (L', V, Ax)$ an (formalized) ontology called a reference ontology for $\mathcal{T}$. An ontological mapping from $\mathcal{T}$ into $\mathcal{O}$ is a (partial) function $f$ from $L$ into $L'$ such that for every concept $c$ in $\GConc$ the expressions $\GDef(c)$ and $f(\GDef(c))$ are semantically equivalent with respect to $Ax$. In this case we may define a formal knowledge base $\GOntoBase(\mathcal{T}) = \{ f(\GDef(c)) \vert c \Lin
\GConc\} \Lunion Ax$ which explicitly extracts the content in $\mathcal{T}$ and provides inference mechanisms. Note, that $\mathcal{O}$ is in general not a foundational ontology, but we assume that $\mathcal{O}$ is constructed from a foundational ontology, say GFO, by a number of well-defined steps. A detailed discussion of this method is presented in (31).

Robert Hoehndorf 2006-10-18


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